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A291026 p-INVERT of the positive integers, where p(S) = 1 - 4*S + S^2. 2
4, 23, 128, 711, 3948, 21920, 121700, 675673, 3751296, 20826953, 115629868, 641969344, 3564171060, 19788040311, 109861881472, 609945846247, 3386378699324, 18800948912352, 104381615697460, 579519775642745, 3217455182279552, 17863096800262569 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Suppose s = (c(0), c(1), c(2), ...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x). Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453).

See A290890 for a guide to related sequences.

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (8, -15, 8, -1)

FORMULA

G.f.: (4 - 9 x + 4 x^2)/(1 - 8 x + 15 x^2 - 8 x^3 + x^4).

a(n) = 8*a(n-1) - 15*a(n-2) + 8*a(n-3) - a(n-4).

MATHEMATICA

z = 60; s = x/(1 - x)^2; p = 1 - 4 s + s^2;

Drop[CoefficientList[Series[s, {x, 0, z}], x], 1] (* A000027 *)

u = Drop[CoefficientList[Series[1/p, {x, 0, z}], x], 1]  (* A291026 *)

CROSSREFS

Cf. A000027, A290890.

Sequence in context: A024050 A236421 A227639 * A239399 A193808 A038723

Adjacent sequences:  A291023 A291024 A291025 * A291027 A291028 A291029

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Aug 19 2017

STATUS

approved

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Last modified January 28 00:32 EST 2020. Contains 331313 sequences. (Running on oeis4.)